In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects [GRAPHICS] with initial data [GRAPHICS] where alpha and nu are positive constants such that alpha < 1, nu < 4 alpha(1 - alpha). Under the assumption that vertical bar psi(+) - psi-vertical bar + vertical bar theta(+) - theta(-)vertical bar is sufficiently small, we show that if the initial data is a small perturbation of the parabolic system defined by (2.4) which are obtained by the convection-diffusion equations (2.1). and solutions to Cauchy problem (E) and (I) tend asymptotically to the convection-diffusion system with exponential rates. Precisely speaking, we derive the asymptotic profile of (E) by Gauss kernel G(t, x) as follows: [GRAPHICS] The same problem was studied by Tang and Zhao [S.Q. Tang, H. J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl. 233 (1999) 336-358], Nishihara [K. Nishihara, Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity, Z. Angew. Math. Phys. 57 (4) (2006) 604-614] for the case of (psi(+/-),theta(+/-)) = (0, 0). (c) 2008 Elsevier Inc. All rights reserved.
